Complex Analysis and Trig

  • Thread starter metgt4
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Homework Statement



The principal valueof the logarithmic function of a complex variable is defined to ave its argument in the range -pi < arg(z) < pi. By writing z = tan(w) in terms of exponentials, show that:

tan-1(z) = (1/2i)ln[(1 + iz)/(1 - iz)]


The Attempt at a Solution



I have absolutely no idea where to start on this problem. My brain must be fried this week, but all I know how to do is write z = tan(w) in terms of exponentials.

z = (1/2i)(eiw + e-iw)/(eiw - eiw)


Thanks!
Andrew
 

Answers and Replies

  • #2
Dick
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First you'd better check your expansion of z in terms of e^(iw) and e^(-iw). There's more than one mistake in there. Once you done that, put x=e^(iw). Then e^(-iw)=1/x. Solving for z in terms of x just means solving a quadratic equation.
 

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